Cornyn said the change would mean no expansion of the so-called Hyde Amendment, which bars annual spending bills from funding abortions except in cases of rape, incest and the health of the mother.

The issue has held up Cornyn’s sex trafficking bill for more than a week, with Democrats balking at any measure blocks access to abortions for victims of sex trafficking crimes.

“What we’re offering here is a middle ground,” Cornyn said.

Democratic leaders seemed little inclined to take the deal, arguing that Cornyn’s new plan would still hinder young women seeking help from the victims’ fund. Those victims, said Vermont Sen. Patrick Leahy, the ranking Democrat on the Judiciary Committee, “should have all options available.”

Still, Republicans need only a handful of Democrats to break ranks and end a weeklong Senate filibuster. Cornyn’s offer came after the third attempt in as many days to cut off debate on the bill. Thursday’s 56-42 “cloture” vote left Republicans just four senators short of the 60 needed to break a filibuster.

Democrats have demanded that Cornyn remove the abortion restrictions from the bill altogether, something the Texas Republican said would be “unacceptable.”

Meanwhile, the debate has all but ended action on any other Senate business, including the confirmation of Attorney General nominee Loretta Lynch, who would be the first African American woman to lead the Justice Department.